I am trying to do Exercises 7.2.3 in the book Graph Theory by Bondy and Murty, which wants the reader to prove that the Ford-Fulkerson Algorithm terminates whenever all capacities are rational.
In their version of the Ford-Fulkerson algorithm, they start with an initial flow that is an arbitrary feasible flow rather than the zero-flow. I know how to prove that this algorithm terminates for rational capacities when the initial flow is also rational, but how can I prove this when the flow is irrational?
You can't: if initial flow is irrational and all initial capacities are rational, then the algorithm doesn't have to terminate.
Take standard example, replace irrational capacity $r$ with some larger rational capacity $q$, and assume initial irrational flow goes through $s \to v_4 \to v_3 \to t$ with capacity $q - r$.